Contour Deformation Method Research Articles

Convex optimization of contour deformations

We discuss various formal aspects of contour deformations used to alleviate sign problems; most importantly, relating these contour deformations to a certain convex optimization problem. As a consequence of this connection we describe a general method for proving upper bounds on the average phase achievable by the contour deformation method. Using this method we show that Abelian lattice Yang-Mills in two spacetime dimensions possesses, for many values of the complex coupling, an exponential sign problem that cannot be removed via any contour deformation. Published by the American Physical Society 2024

Lellouch-Lüscher factor for the K → 3π decays

We derive an explicit expression for the Lellouch-Lüscher (LL) factor in the K → 3π decays at leading order (without derivative couplings). Several important technical details are addressed, like a proper decomposition into the isospin amplitudes, the choice of a minimal set of effective couplings and the renormalization, as well as the algorithm for the solution of the pertinent Faddeev equations in the infinite volume which is based on the contour deformation method. Most importantly, our numerical results demonstrate that the three-body force contributes very little to the LL factor. This result paves the way for the study of the K → 3π decays on the lattice.

How to Determine the Branch Points of Correlation Functions in Euclidean Space II: Three-Point Functions

The analytic structure of elementary correlation functions of a quantum field is relevant for the calculation of masses of bound states and their time-like properties in general. In quantum chromodynamics, the calculation of correlation functions for purely space-like momenta has reached a high level of sophistication, but the calculation at time-like momenta requires refined methods. One of them is the contour deformation method. Here we describe how to employ it for three-point functions. The basic mechanisms are discussed for a scalar theory, but they are the same for more complicated theories and are thus relevant, e.g., for the three-gluon or quark-gluon vertices of quantum chromodynamics. Their inclusion in existing truncation schemes is a crucial step for investigating the analytic structure of elementary correlation functions of quantum chromodynamics and the calculation of its spectrum from them.

Scalable Molecular GW Calculations: Valence and Core Spectra.

We present a scalable implementation of the GW approximation using Gaussian atomic orbitals to study the valence and core ionization spectroscopies of molecules. The implementation of the standard spectral decomposition approach to the screened-Coulomb interaction, as well as a contour-deformation method, is described. We have implemented both of these approaches using the robust variational fitting approximation to the four-center electron repulsion integrals. We have utilized the MINRES solver with the contour-deformation approach to reduce the computational scaling by 1 order of magnitude. A complex heuristic in the quasiparticle equation solver further allows a speed-up of the computation of core and semicore ionization energies. Benchmark tests using the GW100 and CORE65 data sets and the carbon 1s binding energy of the well-studied ethyl trifluoroacetate, or ESCA molecule, were performed to validate the accuracy of our implementation. We also demonstrate and discuss the parallel performance and computational scaling of our implementation using a range of water clusters of increasing size.

Resonance studies using the contour deformation method in the complex momentum plane

A contour deformation method (CDM) in the complex momentum plane has been successfully extended and implemented to probe resonances in atomic and molecular systems. Specifically, solution of the Schrödinger equation is performed in momentum space with momentum deformed on a contour in the complex plane. The bound, resonant, and complex continuum states could be directly revealed from the eigenvalues of the Schrödinger equation in the complex momentum plane. The calculations of shape resonances in electron scattering with Na+ in Debye plasmas (one channel), and in the charge transfer process H−(1s2)+Li(1s22s) (12Σ+) →H(1s)+Li−(1s22s2) (22Σ+) (coupled channels) are given as illustrative examples. It is shown that calculated results from CDM agree very well with those extracted from the eigenphase sum of scattering theories. The effectiveness of CDM is also demonstrated by comparing its results with those obtained by the complex rotation scaling and exterior complex scaling methods. The convergence of CDM results can be obtained by increasing the momentum integration region and the number of integration points. The studied examples demonstrate that CDM could be a powerful tool for studies of resonances in complex atomic and molecular systems.

GW approximation study of the Compton profile of ZnSe

ABSTRACTWe have obtained the Compton profile of ZnSe from the first principles GW approximation (GWA) method and ground-state density functional theory (DFT) method. We observe that between 0 and 1.5 a.u., there is better agreement to previous studies via the GWA difference profile compared to the ground-state difference profile. Above 1.5 a.u., both cases do not agree with the trend of the previous study; however, the application of the GWA is seen to improve the agreement compared to localized density approximation. Previous studies have reported that discrepancies from experiment are related to pseudopotential calculations which have been observed to overestimate momentum density between 0 and 1.5 a.u., while the reverse trend is seen above 1.5 a.u. We thus conclude that improvement to the pseudopotential technique to obtain the Compton profile is possible if the sharp Fermi break of the momentum distribution between high and low momenta becomes more smeared. Using the broadened spectral functions via the contour deformation method to obtain the momentum distributions, the GWA is a natural tool to achieve this via the contribution from the dielectric screening to the quasiparticle energies.

The Contour Deformation Method for Calculating the High-Frequency Scattered Field by the Fock Current on the Surface of the 3-D Convex Cylinder

In this paper, the high-frequency diffracted waves like the creeping waves are comprehensively analyzed by the Fock currents. On invoking the contour deformation method, the highly oscillatory Fock currents are efficiently calculated. Furthermore, the workload for the calculation of Fock currents is frequency-independent. To capture the high-frequency wave physics phenomenon, the Fock current is separated into the classical physical optics (PO) current and the nonuniform (NU)-Fock current along the shadow boundary and in the deep shadow region. To calculate the highly oscillatory scattered wave fields from the Fock current, quadratic approximations of the phase functions in the integrand are adopted. On invoking the numerical steepest descent path (NSDP) method, the scattered wave fields are efficiently calculated with frequency-independent computational effort and error controllable accuracy in each frequency-independent segment. Meanwhile, the high-frequency creeping wave coming from the NU-Fock current is efficiently captured by the NSDP method. Numerical results for the Fock currents, the high-frequency NU-diffracted and scattered far fields on the convex cylinders are given to validate the efficiency of the proposed method. Furthermore, the contour deformation method for computing the Fock currents offers a clear physical picture for the high-frequency wave fields on the convex scatterer.

Study of resonant structures in a deformed mean field by the contour deformation method in momentum space

Solution of the momentum space Schr\"odinger equation in the case of deformed fields is being addressed. In particular it is shown that a complete set of single particle states which includes bound, resonant and complex continuum states may be obtained by the Contour Deformation Method. This generalized basis in the complex energy plane is known as a Berggren basis. The momentum space Schr\"odinger equation is an integral equation which is easily solved by matrix diagonalization routines even for the case of deformed fields. The method is demonstrated for axial symmetry and a fictitious "deformed $^5$He", but may be extended to more general deformation and applied to truly deformed halo nuclei.

The contour deformation method in momentum space, applied to subatomic physics

A generalized contour deformation method (CDM), which combines complex rotation and translation in momentum space, is discussed. CDM gives accurate calculation of two-body spectral structures: bound, antibound, resonant and continuum states forming a Berggren basis. It provides a basis for full off-shell t-matrix calculations both for real and complex input energies. Results for both spectral structures and scattering amplitudes compare perfectly well with exact values for the analytically solvable separable non-local Yamaguchi potential as a testcase. Accurate calculation of antibound states in the Malfliet–Tjon and the realistic CD–Bonn nucleon–nucleon potential are presented. Calculation of antibound states in the CD–Bonn potential are not known to have been given elsewhere.

Optimization Method for Educing Variable-Impedance Liner Properties

A new approach to educing normal incidence acoustic impedance of nonuniform test specimens in grazing incidence and grazing-flow environments is described. An optimization algorithm is shown to provide an efficient means for searching out an impedance distribution to match the measured acoustic field along the upper wall of a duct opposite the test specimen. The flow duct is allowed to transmit multimodal, nonprogressive acoustic waves in a flow environment; however, only a no-flow environment is discussed. A key feature of the method is the expansion of the impedance function as a plecewise continuous polynomial with undetermined coefficients. The upper wall acoustic pressure is computed numerically as a function of these coefficients by using a finite element method. The Davidon-Fletcher-Powell optimization algorithm is used to educe the normal incidence impedance by determining the values of the undetermined coefficients that minimize the difference between the measured and the numerically computed upper wall pressure. Results show that this more robust method reduces by a factor of 30 the time required to make impedance determinations as compared with the contour deformation method and is better suited for liners with spatially varying impedance.

Simple derivation of a general relationship between imaginary- and real-time Green’s/correlation functions

Using a simple contour deformation method it is shown that there is a very general relationship between imaginary-time correlation functions and real-time commutators. This method is used to show easy derivations of the equivalence between (1) two dissimilar forms of the Kubo formula for the electrical conductivity tensor σμν(ω), and (2) the imaginary-time-ordered Matsubara and the real-time retarded Green’s functions. In deriving example (1), the infra- red divergent paramagnetic and diamagnetic parts of σμν(ω) arise naturally from the imaginary-time formulation, and can be explicitly combined to give a single simple regular expression at ω→0.

A three-body calculation for polarization effects in neutron-deuteron scattering

The Faddeev equations with separable two-particle interactions have been solved for the 2n + p system at energies 14.1 MeV and 22.7 MeV respectively. The simultaneous and individual effect of the S- and P-wave interactions and the tensor force on the differential cross section and polarizations has been studied. The experimental data have not been reproduced satisfactorily. A method different from the contour-deformation method has been suggested to treat the singularities of the kernels of the Faddeev integral equations.

Numerical solution of the Faddeev-amado-integral equation without deformation of contours

A method for solving the Faddeev equation in the separable potential approximation is presented in which numerical and analytical integration are mixed to account for the singular structure of the kernel. Numerical results are compared to those obtained by the usual contour deformation method.